Ch. 3, Kinematics in 2 Dimensions; Vectors

Vectors • General discussion.

Vector A quantity with magnitude & direction.

Scalar A quantity with magnitude only.• Here: We mainly deal with

Displacement D & Velocity vOur discussion is valid for any vector!

• The vector part of the chapter has a lot of math! It requires detailed knowledge of trigonometry.

• Problem SolvingA diagram or sketch is helpful & vital! I don’t see how it

is possible to solve a vector problem without a diagram!

Coordinate Systems • Rectangular or Cartesian Coordinates

– “Standard” coordinate axes.

– Point in the plane is (x,y)

– Note, if its convenient

could reverse + & -

- ,+ +,+

- , - + , -

Standard set of xy coordinate axes

Vector & Scalar Quantities

Vector A quantity with magnitude &

direction.

Scalar A quantity with magnitude only.

• Equality of two vectors

2 vectors, A & B. A = B means A & B have the same magnitude & direction.

Sect. 3-2: Vector Addition, Graphical Method • Addition of scalars: “Normal” arithmetic!

• Addition of vectors: Not so simple!

• Vectors in the same direction:– Can also use simple arithmetic

Example: Travel 8 km East on day 1, 6 km East on day 2.

Displacement = 8 km + 6 km = 14 km East

Example: Travel 8 km East on day 1, 6 km West on day 2.

Displacement = 8 km - 6 km = 2 km East

“Resultant” = Displacement

• Adding vectors in the same direction:

Graphical Method • For 2 vectors NOT along same line, adding is

more complicated:

Example: D1 = 10 km East, D2 = 5 km North. What is the resultant (final) displacement?

• 2 methods of vector addition:

– Graphical (2 methods of this also!)

– Analytical (TRIGONOMETRY)

• 2 vectors NOT along same line: D1 = 10 km E, D2 = 5 km N.

Resultant = DR = D1 + D2 = ?

In this special case ONLY, D1 is perpendicular to D2.

So, we can use the Pythagorean Theorem.

Graphical Method: Measure. Find DR = 11.2 km, θ = 27º N of E

= 11.2 km

Note! DR < D1 + D2

(scalar addition)

• Example illustrates general rules (“tail-to-tip” method of

graphical addition). Consider R = A + B

1. Draw A & B to scale.

2. Place tail of B at tip of A

3. Draw arrow from tail of A to tip of BThis arrow is the resultant R (measure length & the angle it

makes with the x-axis)

Order isn’t important! Adding the vectors in the opposite order gives the same result:

In the example, DR = D1 + D2 = D2 + D1

Graphical Method • Adding (3 or more) vectors

V = V1 + V2 + V3

Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

Graphical Method • Second graphical method of adding vectors

(equivalent to the tail-to-tip method!)

V = V1 + V2

1. Draw V1 & V2 to scale from common origin.

2. Construct parallelogram using V1 & V2 as 2 of the 4 sides.

Resultant V = diagonal of parallelogram from common origin (measure length and the angle it makes with the x-axis)

Mathematically, we can move vectors around (preserving their magnitudes & directions)

A common error!

Parallelogram Method

Subtraction of Vectors • First, define the negative of a vector:

- V vector with the same magnitude (size) as V but with opposite direction.

Math: V + (- V) 0

Then add the negative vector.

• For 2 vectors, V1 & V2:

V1 - V2 V1 + (-V2)

Multiplication by a ScalarA vector V can be multiplied by a scalar c

V' = cV

V' vector with magnitude cV & same direction as V

If c is negative, the resultant is in the opposite direction.

Example• A two part car trip. First, displacement A = 20 km

due North. Then, displacement B = 35 km 60º West

of North. Figure. Find (graphically) resultant displacement vector R (magnitude & direction).

R = A + BUse ruler & protractor

to find length of R,

angle β.

Length = 48.2 km

β = 38.9º